The prediction of system responses for a given fatigue test bench drive signal is a challenging task, for which linear frequency response function models are commonly used. To account for non-linear phenomena, a novel hybrid model is suggested, which augments existing approaches using Long Short-Term Memory networks. Additional virtual sensing applications of this method are demonstrated. The approach is tested using non-linear experimental data from a servo-hydraulic test rig and this dataset is made publicly available. A variety of metrics in time and frequency domains, as well as fatigue strength under variable amplitudes, are employed in the evaluation.